For the first systematic investigations of the theory of cluster sets of analytic functions, we are indebted to IVERSEN 1-3J and GROSS 1-3J about forty years ago. Subsequent important contributions before 1940 were made by SEIDEL 1-2J, DOOE 1-4J, CARTWRIGHT 1-3J and BEURLING 1]. The investigations of SEIDEL and BEURLING gave great impetus and interest to Japanese mathematicians; beginning about 1940 some contributions were made to the theory by KUNUGUI 1-3J, IRIE IJ, TOKI IJ, TUMURA 1-2J, KAMETANI 1-4J, TsuJI 4J and NOSHIRO 1-4J. Recently, many noteworthy advances have been made...